Fermionic R-Operator for the Fermion Chain Model.pdf

a r X i v : h e p - t h / 9 8 0 6 0 8 3 v 1 1 1 J u n 1 9 9 8 Fermionic R-Operator for the Fermion Chain Model Yukiko Umeno ?, Masahiro Shiroishi ?? and Miki Wadati Department of Physics, Graduate School of Science, University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo 113-0033 (Received December 15, 1997) The integrability of the one-dimensional (1D) fermion chain model is investigated in the framework of the Quantum Inverse Scattering Method (QISM). We introduce a new R- operator for the fermion chain model, which is expressed in terms of the fermion operators. The R-operator satisfies a new type of the Yang-Baxter relation with fermionic L-operator. We derive the fermionic Sutherland equation from the relation, which is equivalent to the fermionic Lax equation. It also provides a mathematical foundation of the boost operator approach for the fermion model. In fact, we obtain some higher conserved quantities of the fermion model using the boost operator. KEYWORDS: quantum inverse scattering method, fermionic formulation, fermion chain model, Sutherland equation, boost operator 1 Introduction In the last decades, many integrable spin chain models have been investigated by means of Quantum Inverse Scattering Method (QISM for brevity). [1] In the QISM, the R-matrix, which satisfies the Yang-Baxter relation with the L-operator R12(u1, u2) 1 Lj(u1) 2 Lj(u2) = 2 Lj(u2) 1 Lj(u1)R12(u1, u2), (1) plays the essential role. [2, 3, 4, 5, 6] For the spin chain models, the R-matrix R12(u1, u2) is a c-number matrix and satisfies the Yang-Baxter equation R12(u1, u2)R13(u1, u3)R23(u2, u3) = R23(u2, u3)R13(u1, u3)R12(u1, u2). (2) ?E-mail: umeno@monet.phys.s.u-tokyo.ac.jp ??E-mail: siroisi@monet.phys.s.u-tokyo.ac.jp 1 From the Yang-Baxter relation (1), it follows that the transfer matrix defined by t(u) = tra( a LN(u) . . . a L1(u)) (3) constitutes a commuting family [t(u), t(v)] = 0. (4) Expansion of t(u) in terms of spectral parameter u gives the conserved quantities. There- for